Next, we review some important properties of the surplus function E. It is convex and differentiable [1]}. The well-known Williams-Daly-Zachary theorem states that the gradient of E corresponds to the vector of choice probabilities [2]}, i. e. each component gives the probability that alternative \(i \)  provides the maximum utility among all alternatives.
To see this, let us denote the choice probabilities by
\(\mathbb {P}^{(i)} = \mathbb {P} \left( v^{(i)} + \epsilon ^{(i)} = \underset{1 \le i \le n}{\max } v^{(i)} + \epsilon ^{(i)} \right).\) 
